The theory of differential equations in abstract spaces is a fascinating field with important applications to a number of areas of analysis and other branches of mathematics. At the present time, there is no single book that is self-contained and simple enough to appeal to the beginner. Furthermore, if one desires to give a course so as to expose the student to this branch of research, such a book becomes handy. This being the motivation, the aim of our book is, in fact, to introduce the nonspecialist to this elegant theory and powerful techniques. But for some familiarity with the elements of functional analysis, all the important results used in this book are carefully stated in the appendixes so that, for the most part, no other references are needed. The required theory, from the calculus of abstract functions and the theory of semigroups of operators, used in connection with differential equations in Banach spaces is treated in detail.